OSR
On some interesting Curves...
Aditya Kiran
Grad 1st yr
Applied Math
Curvature
• The fact of being curved or the degree to
which something is curved..
• Tendency to deviate from moving in a straight
line..
How to find curvature?
 y =f(x) form
Parametric form
y =f(t) , x=f(t)
 Polar form
(r,theta)
4 ‘interesting’ curves
• We discuss about 4 curves today:
•
•
•
•
1)Catenary
2)Tractrix
3)Archimedean Spiral
4) Tautochrone
#1
Catenary
• Àlso called Alysoid or chain equation..
• The word catenary is derived from the Latin
word ‘catena’ for chain.
• Solved by Huygens (wave theory of light- guy)
equation
• Y = Coshx(x/a)
• Parametric:
• X(t)=t,
• Y(t)=aCosh(t/a)
• Curvature:
• k(t)=
Special properties..
• A catenary is also the locus of the focus of a
parabola rolling on a straight line.
• The catenary is the evolute of the tractrix.
#2
Tractrix
• From the Latin ‘trahere’ meaning drag/pull..
• Also called drag curve or donkey curve.
• The tractrix was first studied by Huygens in
1692
So what’s the problem about?
“What is the path of an object when it is dragged
along by a string of constant length being pulled
along a straight horizontal line”
The Equation :
• Y=a*Sech(t)
• X=a*(t-Tanh(t))
Curvature:
k(t)=Cosech(t)/a
Special properties..
• Tractrix is also obtained when a hyperbolic
spiral is rolled on a straight line..Then locus of
the center of the hyperbola forms a tractrix.
• Evolute of a tractrix is a catenary
• Example in daily life:
..If Car front wheels travel along straight
line, the back wheels follow a tractrix.
#3
Archimedean spiral
• It is the locus of a point moving away from a
fixed point with a constant speed along a line
which rotates with constant angular velocity.
Rotate around
a point with
constant
angular velocity
Move away
from the point
with constant
speed
=
Archimedean
spiral
• Equation in polar coordinates:
• a turns the spiral
• B controls the distance between successive
turnings.
equation
• r= bϴ
• Parametric:
x= b*ϴ*Cos(ϴ)
y= b*ϴ*Sin(ϴ)
• Curvature:
• Any ray from origin meets successive turnings at a
constant separation.
• Used to convert circular motion to linear
motion..
• Used in Archimedes screw
General formula of spiral
• X=1, Archimedean spiral
• X=2 ,Fermat spiral
• X=-1 ,Hyperbolic spiral
• Logarithmic spiral
Logarithmic spiral
#4
Tautochrone
• A tautochrone or isochrone curve
• Tauto=same, and chrono =time
• Solved by Christian Huygens in 1659.
• “the curve for which the time taken by an object
sliding without friction in uniform gravity to its
lowest point is independent of its starting point”
• Its nothing but a cycloid.
• (
• Curvature: k=Cosec(ϴ/2)/4a
• T=
Thank you.!!!